QPSK equation, wave forms and Signal space diagram

QPSK equation:-

The meaning of QPSK is that the carrier signal takes on different phases Π/4, 3Π/4, 5Π/4 and 7Π/4 based on incoming di-bit combination or symbol.

$\large S_{QPSK}(t)= \sqrt{\frac{2E_{s}}{T_{s}}}cos(2\pi f_{c}t +(2i-1)\frac{\pi}{4}), 0\leq t\leq T_{s}$

= 0, elsewhere, where i = 1,2,3,4.

E_b and T_b are the bit energy and bit-interval , E_s and T_s are the energy per symbol and symbol duration. T_s= 2 T_b .

The carrier frequency f_c = n_c /T_s. where n_c is a fixed integer.

each possible value of phase corresponds to a unique di-bit. then the foregoing phase values to represent the gray encoded set of di-bits 11,01,10 and 00, where only a single bit is changed from one di-bit to the next.

QPSK equation can be represented in another format as follows

$\large S_{QPSK}(t) = \sqrt{\frac{2E_{s}}{T_{s}}}cos (2\pi f_{c}t+(2i+1)\frac{\pi }{4} ), 0\leq t\leq T_{s}$

= 0, elsewhere ,where i=0,1,2,3.

The above two equations are same, there is a change in i values. alternately the equation can be represented as follows.

$S_{QPSK}(t)= \sqrt{\frac{2E_{s}}{T_{s}}}cos (2i-1)\frac{\pi }{4}cos2\pi f_{c}t - \sqrt{\frac{2E_{s}}{T_{s}}}sin (2i-1)\frac{\pi }{4}sin2\pi f_{c}t$ , where i= 1,2,3,4.